All math should start in the concrete. Fractions are represented by real objects cut into equal portions. Addition adds blocks together, subtraction takes them apart. Multiplication has flowers that need to be put into vases. Division has legos that need to be distributed evenly. These concepts start long before anything is put onto paper. Kindergartners, most of the time, do not learn formal division but they should know how to put things into groups.
Math should be taught in an incremental way. Although many old (and current) textbooks are written with blocks of addition followed by blocks of subtraction, students should learn them in a formal way (semi-concrete through abstract learning) together. Especially considering older students building on what they've already learned, math should be taught in bits and pieces and constantly reviewed.
Math should not be a series of rote problems followed by the dreaded word problems. Problem solving should come first, with the rote method (computation) following in order to solve problems in an orderly way. Like rote skills, problem solving must be practiced.. The goal of a math program is not to get a better score on a standardized test. The goal of a math program is to be able to use math outside of the school day. Being able to do arithmetic is of little value if one does not know when to use the skills. Real-life situations (“when will I ever have to use this?”) must be the basis of all math teaching.
Math is truly outcome-based. Understanding comes incrementally and builds upon itself. Again, the end goal is not an arbitrary score or grade, but mathematical thinking. In order to create mathematical thinking, a teacher must be able to think mathematically herself. She must also, however, remain in a state of knowing how she knows something. Consider learning in four stages:
Teachers must remain in the “Practice” stage of learning even if they are masters of a subject (hopefully they are!). The worst thing a child—or adult student—can encounter in a math classroom is a teacher who is such an expert in the field that she can no longer explain how she knows something. It has become part of her being and she stands there in front of her classroom, not understanding why her students don't “get it.” Good teachers know they must shift down a notch to the practice stage of learning and keep in their minds, always, how it was that they first understood this concept.
This is especially true in math teaching. Since math is taught incrementally and beginning in the most concrete concepts, teachers must be careful not to teach the shortcut or the rote skill in isolation from the mathematical thinking. Explaining to students that when dividing by a fraction, “just flip the second fraction and multiply”, while completely factual, skips over all the reasons behind why it is that this is true. It makes math a mystery full of tricks. Instead, beginning with concrete concepts, testing them out, and then introducing the rules with explanation, fashions math into a tool.
[Side note: how would I teach division of fractions? I would first teach multiplication of fractions and how multiplying by ¼ is the same as dividing by 4, with many iterations of this idea. I would bring in something, say a block of cheese, that weighed a half pound, and have students divide it by 3. This is the same as cutting it into thirds (multiplying by 1/3). I would have them weigh each piece and find the answer. I'd have a loaf of bread 8 inches long that needs to be cut into 2/3 inch slices. How many slices? And so forth. Using the data from these concrete examples, I would move to the algorithm of writing 8 / 2/3. Then I would ask (since they would already understand common denominators before this lesson): how many thirds-of-an-inch is 8 inches? It is 24 thirds-of-an-inch. Now I want 2/3 of an inch slices: 24 divided by 2 is 12. Problems would continue in this style until the student masters the why: at that point, the math shortcut would be introduced if it hadn't been derived by the student already: if you simply find the reciprocal of 2/3, which is 3/2, and then multiply by 8, the answer is 24/2. Simplified, that is 12.]
Math require practice. Concepts may come easily to some students but skills require practice. Rote practice of both facts and higher skills is necessary to keep a student from becoming bogged down in computation. Not every student needs extensive skills practice, but students should demonstrate progress towards mastery of computational skills before moving forward with higher concepts. There are exceptions to this general rule, of course. Students with certain learning differences and disabilities find memorization of math facts cumbersome and sometimes impossible. It is a disservice to keep these students from learning higher math concepts when technology can aid them in computation. This should not be the first route taken, but should be in the teacher's arsenal of adaptations. Most upper level math courses assume a calculator, even beginning in middle school. Students should understand the concepts but if the computation gets in the way, they should be allowed a bridge across that chasm after careful consideration.
Mathematics is a natural process. Young children learn quickly how to create “fair” portions of grapes and cookies. Counting steps, fingers, plates on the table, and so forth are obvious first steps in math. As opposed to reading, which is not an innate process but something we have trained our brains to do over the centuries, math has always been here. Moving through a math curriculum in the right way, with incremental steps, concrete to abstract, reviewing and practicing with a teacher who understands how one learns these concepts, creates mathematical thinkers who know exactly when they will “have to use this in real life.”